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Self Portrait (lithograph, 1948) His interest began in 1936, when he traveled to Spain and viewed the tile patterns used in the Alhambra. He spent many days sketching these tilings, and later claimed that this “was the richest source of inspiration that I have ever tapped.” In 1957 he wrote an essay on tessellations, in which he remarked: In mathematical quarters, the regular division of the plane has been considered theoretically . . . Does this mean that it is an exclusively mathematical question? In my opinion, it does not. [Mathematicians] have opened the gate leading to an extensive domain, but they have not entered this domain themselves. By their very nature they are more interested in the way in which the gate is opened than in the garden lying behind it. Whether or not this is fair to the mathematicians, it is true that they had shown that of all the regular polygons, only the triangle, square, and hexagon can be used for a tessellation. He also elaborated these patterns by “distorting” the basic shapes to render them into animals, birds, and other figures. These distortions had to obey the three, four, or six-fold symmetry of the underlying pattern in order to preserve the tessellation. Now in Leonardo's early writings we find him echoing the precise theory of perspective as set out by Alberti and Piero. He developed mathematical formulas to compute the relationship between the distance from the eye to the object and its size on the intersecting plane, that is the canvas on which the picture will be painted:- If you place the intersection one metre from the eye, the first object, being four metres from the eye, will diminish by three-quarters of its height on the intersection; and if it is eight metres from the eye it will diminish by seven-eighths and if it is sixteen metres away it will diminish by fifteen-sixteenths, and so on. As the distance doubles so the diminution will double. Other artists would go on to further define perspective and the mathematical relationships within art. MATH ART IDEAS Leonardo da Vinci perspective drawing. This article looks at some of the interactions between mathematics and art in western culture. There are other topics which will look at the interaction between mathematics and art in other cultures. Before beginning the discussion of perspective in western art, we should mention the contribution by al-Haytham. It was al-Haytham around 1000 A.D. who gave the first correct explanation of vision, showing that light is reflected from an object into the eye. He studied the complete science of vision, called perspectiva in medieval times, and although he did not apply his ideas to painting, the Renaissance artists later made important use of al-Haytham's optics. There is little doubt that a study of the development of ideas relating to perspective would be expected to begin with classical times, and in particular with the ancient Greeks who used some notion of perspective in their architecture and design of stage sets. However, although Hellenistic painters could create an illusion of depth in their works, there is no evidence that they understood the precise mathematical laws which govern correct representation. We chose to begin this article, therefore, with the developments in the understanding of perspective which took place during the Renaissance. First let us state the problem: how does one represent the three-dimensional world on a two-dimensional canvass? There are two aspects to the problem, namely how does one use mathematics to make realistic paintings and secondly what is the impact of the ideas for the study of geometry. Maurits Cornelis Escher, who was born in Leeuwarden, Holland in 1898, created unique and fascinating works of art that explore and exhibit a wide range of mathematical ideas. While he was still in school his family planned for him to follow his father's career of architecture, but poor grades and an aptitude for drawing and design eventually led him to a career in the graphic arts. His work went almost unnoticed until the 1950’s, but by 1956 he had given his first important exhibition, was written up in Time magazine, and acquired a world-wide reputation. Among his greatest admirers were mathematicians, who recognized in his work an extraordinary visualization of mathematical principles. This was the more remarkable in that Escher had no formal mathematics training beyond secondary school. As his work developed, he drew great inspiration from the mathematical ideas he read about, often working directly from structures in plane and projective geometry, and eventually capturing the essence of non-Euclidean geometries. Reptiles (62k) Regular Divisionof the Planewith Birds (21k) Development 1 (59k) Cycles (40k) http://www.mathacademy.com/pr/minitext/escher/ http://www.mathartfun.com/shopsite_sc/store/html/index.html http://www-history.mcs.st-and.ac.uk/HistTopics/Art.html http://www.segerman.org/tshirts.html The works of MC Escher. http://eschermathfall2009.umwblogs.org/2009/10/14/project-1-houston-we-have-a-tessellation-by-devin-netter/ http://maa.missouri.edu/pdfs/LessonMathInArt.doc
Certainly these books are not simply the same work translated into two different languages. Rather Alberti addresses the books to different audiences, the Latin book is much more technical and addressed to scholars while his Italian version is aimed at a general audience. De pictura is in three parts, the first of which gives the mathematical description of perspective which Alberti considers necessary to a proper understanding of painting. It is, Alberti writes:- ... completely mathematical, concerning the roots in nature from which arise this graceful and noble art. In fact he gives a definition of a painting which shows just how fundamental he considers the notion of perspective to be:- A painting is the intersection of a visual pyramid at a given distance, with a fixed centre and a defined position of light, represented by art with lines and colours on a given surface. Alberti gives background on the principles of geometry, and on the science of optics. He then sets up a system of triangles between the eye and the object viewed which define the visual pyramid referred to above. He gives a precise concept of proportionality which determines the apparent size of an object in the picture relative to its actual size and distance from the observer. One of the most famous examples used by Alberti in his text was that of a floor covered with square tiles. For simplicity we take the centric point, as Alberti calls it (today it is called the vanishing point), in the centre of the square picture. The most mathematical of all the works on perspective written by the Italian Renaissance artists in the middle of the 15th century was by Piero della Francesca. In some sense this is not surprising since as well as being one of the leading artists of the period, he was also the leading mathematician writing some fine mathematical texts. In Trattato d'abaco which he probably wrote around 1450, Piero includes material on arithmetic and algebra and a long section on geometry which was very unusual for such texts at the time. It also contains original mathematical results which again is very unusual in a book written in the style of a teaching text (although in the introduction Piero does say that he wrote the book at the request of his patron and friends and not as a school book). Is there a connection with perspective? Yes there is, for Piero illustrates the text with diagrams of solid figures drawn in perspective.We see from this introduction that Piero intends to concentrate on the mathematical principles. Perhaps it is most accurate to say that he is studying the geometry of vision which he later makes clearer:- First is sight, that is to say the eye; second is the form of the thing seen; third is the distance from the eye to the thing seen; fourth are the lines which leave the boundaries of the object and come to the eye; fifth is the intersection, which comes between the eye and the thing seen, and on which it is intended to record the object. The person who is credited with the first correct formulation of linear perspective is Brunelleschi. He appears to have made the discovery in about 1413. He understood that there should be a single vanishing point to which all parallel lines in a plane, other than the plane of the canvas, converge. Also important was his understanding of scale, and he correctly computed the relation between the actual length of an object and its length in the picture depending on its distance behind the plane of the canvas. Using these mathematical principles, he drew two demonstration pictures of Florence on wooden panels with correct perspective. One was of the octagonal baptistery of St John, the other of the Palazzo de Signori. To give a more vivid demonstration of the accuracy of his painting, he bored a small hole in the panel with the baptistery painting at the vanishing point. A spectator was asked to look through the hole from behind the panel at a mirror which reflected the panel. In this way Brunelleschi controlled precisely the position of the spectator so that the geometry was guaranteed to be correct. These perspective paintings by Brunelleschi have since been lost but a "Trinity" fresco by Masaccio from this same period still exists which uses Brunelleschi's mathematical principles. Piero's illustration of a dodecahedron Masaccio's Holy Trinity Now although it is clear that Brunelleschi understood the mathematical rules involving the vanishing point that we have described above, he did not write down an explanation of how the rules of perspective work. The first person to do that was Alberti in his treatise On painting. Now in fact Alberti wrote two treatises, the first was written in Latin in 1435 and entitled De pictura while the second, dedicated to Brunelleschi, was an Italian work written in the following year entitled Della pittura. Alberti's construction of perspective for a tiled floor A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale. There are many mathematical structures that are fractals; e.g. Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor. In our diagram the centric point is C. The square tiles are assumed to have one edge parallel to the bottom of the picture. The other edges which in reality are perpendicular to these edges, will appear in the picture to converge to the centric point C. The diagonals of the squares will all converge to a point D on a line through the centric point parallel to the bottom of the picture. The length of CD determines the correct viewing distance, that is the distance the observer has to be from the picture to obtain the correct perspective effect. Jock Cooper's Fractal Art https://www.fractalus.com/fractal-art-faq/faq03.html http://www.fractal-recursions.com/ http://www-history.mcs.st-and.ac.uk/HistTopics/Art.html
